Integration by Parts
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Integral Calculus

86223 \(\int \frac{e^{x}(1+\sin x)}{1+\cos x} d x=\)

1 \(e^x \tan \left(\frac{x}{2}\right)+C\)
2 \(e^x \tan x+C\)
3 \(e^x\left(\frac{1+\sin x}{1-\cos x}\right)+C\)
4 \(C-e^x \cot \left(\frac{x}{2}\right)\)
Karnataka CET-2015
Integral Calculus

86225 If
\(\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=\operatorname{alog}\left|\frac{x-1}{x+1}\right|+b \tan ^{-1}\left(\frac{x}{2}\right)+C\) then

1 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{-1}{2}\)
2 \(\mathrm{a}=1, \mathrm{~b}=-1\)
3 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}\)
4 \(\mathrm{a}=-1, \mathrm{~b}=1\)
MHT CET-2020
Integral Calculus

86226 \(\int \frac{x+1}{x^{2}+5 x+6} d x=\)

1 \(-\log |\mathrm{x}+2|+2 \log |\mathrm{x}+3|+\mathrm{c}\)
2 \(\log |x+2|+2 \log |x+3|+c\)
3 \(-\log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
4 \(2 \log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86228 \(\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}^{2}}\right)^{2} \mathrm{dx}=\)

1 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
2 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
3 \(\mathrm{e}^{\mathrm{x}}\left(\frac{1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
4 \(\mathrm{e}^{\mathrm{x}}\left(\frac{2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
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Integral Calculus

86223 \(\int \frac{e^{x}(1+\sin x)}{1+\cos x} d x=\)

1 \(e^x \tan \left(\frac{x}{2}\right)+C\)
2 \(e^x \tan x+C\)
3 \(e^x\left(\frac{1+\sin x}{1-\cos x}\right)+C\)
4 \(C-e^x \cot \left(\frac{x}{2}\right)\)
Karnataka CET-2015
Integral Calculus

86225 If
\(\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=\operatorname{alog}\left|\frac{x-1}{x+1}\right|+b \tan ^{-1}\left(\frac{x}{2}\right)+C\) then

1 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{-1}{2}\)
2 \(\mathrm{a}=1, \mathrm{~b}=-1\)
3 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}\)
4 \(\mathrm{a}=-1, \mathrm{~b}=1\)
MHT CET-2020
Integral Calculus

86226 \(\int \frac{x+1}{x^{2}+5 x+6} d x=\)

1 \(-\log |\mathrm{x}+2|+2 \log |\mathrm{x}+3|+\mathrm{c}\)
2 \(\log |x+2|+2 \log |x+3|+c\)
3 \(-\log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
4 \(2 \log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86228 \(\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}^{2}}\right)^{2} \mathrm{dx}=\)

1 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
2 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
3 \(\mathrm{e}^{\mathrm{x}}\left(\frac{1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
4 \(\mathrm{e}^{\mathrm{x}}\left(\frac{2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
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Integral Calculus

86223 \(\int \frac{e^{x}(1+\sin x)}{1+\cos x} d x=\)

1 \(e^x \tan \left(\frac{x}{2}\right)+C\)
2 \(e^x \tan x+C\)
3 \(e^x\left(\frac{1+\sin x}{1-\cos x}\right)+C\)
4 \(C-e^x \cot \left(\frac{x}{2}\right)\)
Karnataka CET-2015
Integral Calculus

86225 If
\(\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=\operatorname{alog}\left|\frac{x-1}{x+1}\right|+b \tan ^{-1}\left(\frac{x}{2}\right)+C\) then

1 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{-1}{2}\)
2 \(\mathrm{a}=1, \mathrm{~b}=-1\)
3 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}\)
4 \(\mathrm{a}=-1, \mathrm{~b}=1\)
MHT CET-2020
Integral Calculus

86226 \(\int \frac{x+1}{x^{2}+5 x+6} d x=\)

1 \(-\log |\mathrm{x}+2|+2 \log |\mathrm{x}+3|+\mathrm{c}\)
2 \(\log |x+2|+2 \log |x+3|+c\)
3 \(-\log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
4 \(2 \log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86228 \(\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}^{2}}\right)^{2} \mathrm{dx}=\)

1 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
2 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
3 \(\mathrm{e}^{\mathrm{x}}\left(\frac{1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
4 \(\mathrm{e}^{\mathrm{x}}\left(\frac{2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
MHT CET-2020
NEET DIGITAL LIBRARY
Integral Calculus

86223 \(\int \frac{e^{x}(1+\sin x)}{1+\cos x} d x=\)

1 \(e^x \tan \left(\frac{x}{2}\right)+C\)
2 \(e^x \tan x+C\)
3 \(e^x\left(\frac{1+\sin x}{1-\cos x}\right)+C\)
4 \(C-e^x \cot \left(\frac{x}{2}\right)\)
Karnataka CET-2015
Integral Calculus

86225 If
\(\int \frac{2 x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}+4\right)} d x=\operatorname{alog}\left|\frac{x-1}{x+1}\right|+b \tan ^{-1}\left(\frac{x}{2}\right)+C\) then

1 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{-1}{2}\)
2 \(\mathrm{a}=1, \mathrm{~b}=-1\)
3 \(\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}\)
4 \(\mathrm{a}=-1, \mathrm{~b}=1\)
MHT CET-2020
Integral Calculus

86226 \(\int \frac{x+1}{x^{2}+5 x+6} d x=\)

1 \(-\log |\mathrm{x}+2|+2 \log |\mathrm{x}+3|+\mathrm{c}\)
2 \(\log |x+2|+2 \log |x+3|+c\)
3 \(-\log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
4 \(2 \log |\mathrm{x}+2|-2 \log |\mathrm{x}+3|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86228 \(\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}^{2}}\right)^{2} \mathrm{dx}=\)

1 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
2 \(\mathrm{e}^{\mathrm{x}}\left(\frac{-2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
3 \(\mathrm{e}^{\mathrm{x}}\left(\frac{1}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
4 \(\mathrm{e}^{\mathrm{x}}\left(\frac{2}{1+\mathrm{x}^{2}}\right)+\mathrm{C}\)
MHT CET-2020
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